 Surface area and volume of different Geometrical Figures

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Surface area and volume of different Geometrical Figures
Cube Parallelopiped Cylinder Cone

Faces of cube face 1 3 2 Dice (Pasa)
Total faces = 6 ( Here three faces are visible)

Faces of Parallelopiped
Total faces = 6 ( Here only three faces are visible.) Book Brick

Cores Cores Total cores = 12 ( Here only 9 cores are visible)
Note Same is in the case in parallelopiped.

(Here all the faces are rectangular) (Here all the faces are square)
Surface area Cube Parallelopiped c a b a a Click to see the faces of parallelopiped. a (Here all the faces are rectangular) (Here all the faces are square) Surface area = Area of all six faces = 6a2 Surface area = Area of all six faces = 2(axb + bxc +cxa)

Volume of Parallelopiped
Click to animate b c b a Area of base (square) = a x b Height of cube = c Volume of cube = Area of base x height = (a x b) x c

Area of base (square) = a2
Volume of Cube Click to see a a Area of base (square) = a2 Height of cube = a Volume of cube = Area of base x height = a2 x a = a3 (unit)3

Outer Curved Surface area of cylinder
Click to animate Activity -: Keep bangles of same radius one over another. It will form a cylinder. h r r Circumference of circle = 2 π r Formation of Cylinder by bangles It is the area covered by the outer surface of a cylinder. Circumference of circle = 2 π r Area covered by cylinder = Surface area of of cylinder = (2 π r) x( h)

Total Surface area of a solid cylinder
circular surfaces Curved surface = Area of curved surface + area of two circular surfaces =(2 π r) x( h) + 2 π r2 = 2 π r( h+ r)

Other method of Finding Surface area of cylinder with the help of paper
Surface area of cylinder = Area of rectangle= 2 πrh

Volume of cylinder = Area of base x vertical height = π r2 xh

Cone Base r h l = Slant height

Volume of a Cone Click to See the experiment h h
Here the vertical height and radius of cylinder & cone are same. r r 3( volume of cone) = volume of cylinder 3( V ) = π r2h V = 1/3 π r2h

if both cylinder and cone have same height and radius then volume of a cylinder is three times the volume of a cone , Volume = 3V Volume =V

Mr. Mohan has only a little jar of juice he wants to distribute it to his three friends. This time he choose the cone shaped glass so that quantity of juice seem to appreciable.

Surface area of cone l 2πr l l
Area of a circle having sector (circumference) 2π l = π l 2 Area of circle having circumference 1 = π l 2/ 2 π l So area of sector having sector 2 π r = (π l 2/ 2 π l )x 2 π r = π rl

6a2 2π rh π r l 4 π r2 a3 π r2h 1/3π r2h 4/3 π r3 Surface area Volume
Comparison of Area and volume of different geometrical figures Surface area 6a2 2π rh π r l 4 π r2 Volume a3 π r2h 1/3π r2h 4/3 π r3

6r2 =2 π r2 (about) 2π r2 2 π r2 r3 3.14 r3 0.57π r3 0.47π r3 r r/√2 r
Area and volume of different geometrical figures r l=2r r r/√2 r Surface area 6r2 =2 π r2 (about) 2π r2 2 π r2 Volume r3 3.14 r3 0.57π r3 0.47π r3

4π r2 4 π r2 2.99r3 3.14 r3 2.95 r3 4.18 r3 r r Total Surface area r
Total surface Area and volume of different geometrical figures and nature r l=3r r r 1.44r 22r Total Surface area 4π r2 4 π r2 Volume 2.99r3 3.14 r3 2.95 r3 4.18 r3 So for a given total surface area the volume of sphere is maximum. Generally most of the fruits in the nature are spherical in nature because it enables them to occupy less space but contains big amount of eating material.

Think :- Which shape (cone or cylindrical) is better for collecting resin from the tree
Click the next

Long but Light in weight
r r 3r V= 1/3π r2(3r) V= π r3 Long but Light in weight Small niddle will require to stick it in the tree,so little harm in tree V= π r2 (3r) V= 3 π r3 Long but Heavy in weight Long niddle will require to stick it in the tree,so much harm in tree

Cone shape Cylindrical shape Bottle

If we make a cone having radius and height equal to the radius of sphere. Then a water filled cone can fill the sphere in 4 times. V1 r r r V=1/3 πr2h If h = r then V=1/3 πr3 V1 = 4V = 4(1/3 πr3) = 4/3 πr3

Volume of a Sphere Click to See the experiment h=r r
Here the vertical height and radius of cone are same as radius of sphere. 4( volume of cone) = volume of Sphere 4( 1/3πr2h ) = 4( 1/3πr3 ) = V V = 4/3 π r3

Thanks U.C. Pandey R.C.Rauthan, G.C.Kandpal